SCIENCE 



NEW YORK, OCTOBER 2, 1891. 



THE EVOLUTION OF ALGEBRA.' 



In considering the possible subjects for an address on ibis 

 occasion, it has seemed to me that a half-hour might be 

 agreeably spent in a brief survey of the progress, or evolu- 

 tion, of algebra from its earliest known beginnings to the 

 present time. 



The realm of mathematics may be classiSed, in a general 

 way, into (1) Arithmetic, or the theory of numbers, (2) 

 Algebra, (3) Geometry, though sharp dividing lines cannot 

 always be drawn between these departments: the last two, 

 for instance, mutually interacting, geometry illustrating 

 algebra, while algebra is the efficient servant of geometry, 

 enabling it to conquer territory which it could scarcely have 

 entered upon unaided. 



The history of the develbpment of these different branches 

 of mathematics shows considerable diversities among them. 

 Thus geometry reached in a short time, among the ancient 

 Greeks, a high stage of advancement, and then became 

 practically stationary until quite recent times, while the 

 progress of algebra has been more in the nature of a gradual 

 and continuous evolution. Nesselmann has recognized three 

 stages in this development, which he designates as the 

 rhetorical, the syncopated, and the symbolical, to which I 

 may perhaps venture to add the "multiple," in which a 

 plurality of fundamental units is recognized and treated. 

 We may resrard the 6rst three as somewhat analogous to the 

 stone, bronze, and iron ages in human history, overlapping 

 each other, as do these, at different times and places; while 

 the last may be compared to that age of aluminum which is 

 perhaps dawning upon the world. 



Rhetorical algebra was a process for determining the un- 

 known quantity' in an equation by a course of logical reason- 

 ing expressed entirely in words, without the use of any 

 symbols whatever, similar to our present mental arithmetic. 

 In course of time abbreviations of those words which con- 

 stantly recurred were introduced, by the use of which the 

 statement of the reasoning could be much shortened, it being 

 even possible with the notation of Diophantos to approximate 

 to the conciseness of the modern, or symbolic, method. This, 

 however, was not done by Diophantos himself, who used his 

 abbreviations strictly as such, and reasoned out his results in 

 words combined with these. This method is what is desig- 

 nated by Nesselmann as the syncopated, and forms evidently 

 a stepping-stone toward the symbolic, in which perfectly 

 arbitrary symbols are employed to represent the various 

 quantities dealt with, and no words are written out except a 

 conjunction now and then. 



The earliest traces of algebraic knowledge which have been 



discovered are found in Egypt, that wonderful land whose 



records carry us back to such a remote antiquity. Ahmes, 



in a papyrus manuscript, dating from about 1400 B.C., deals 



with certain geometric and algebraic problems, and seems to 



have had as good a conception of the symbolism of algebra 



as his successors of a much later period. Thus he had signs 



1 Address before the Section of Mathematics and Astronomy of the Ameri- 

 can Association for the Advancement of Science, at Washington, D.C., Aug. 

 19-25, 1S91, by E. W. Hyde, vice-president of the section. 



for -|-, — , and ^, and used the character representing a 

 heap for the unknown quantity. He seems, therefore, to 

 have long anticipated Diophantos in the use of syncopated 

 notation. Our knowledge of Egyptian mathematics subse- 

 quent to this time is very slight, and is gleaned from the 

 statements of various Greek and Latin authors. 



We will pass, then, at once to the Greek contributions to 

 the development of our subject. So far as can now be ascer- 

 tained, probably but little strictly algebraic work was done 

 before the third or fourth century of our era, though opinions 

 difPer on this point. The wonderful accomplishments of 

 Archimedes were mainly geometrical and mechanical, though 

 he makes one remark which is equivalent to a statement re- 

 garding the roots of an equation of the third degree, which 

 is remarkable as being, with one exception, the only known 

 case of any consideration of such an equation until after the 

 lapse of more than a thousand years from his time. 



Thymaridas in the second century of our era is the earliest 

 mathematician known to have enunciated an algebraic 

 theorem. This was, however, done entirely in words, no 

 symbol for any quantity or operation being used. 



Practically the foundation of algebra was laid by Dio- 

 phantos of Alexandria. But little is known of this remarka- 

 ble man. Though we have his writings in Greek, he was 

 probably not himself a Greek. The period at which he lived 

 is in dispute, though probabilities favor the fourth century 

 of our era. Even the spelling of his name is uncertain, there 

 being a question as to whether the last syllable should be os 

 or es. But whatever may be known or unknown about the 

 man himself, his writings show a very wonderful power of 

 analytic reasoning, especially when we consider the awkward- 

 ness of the tools with which he was obliged to work. 



What strikes us at once, from our present point of view, 

 as most hampering is the fact that he had only one symbol 

 for the unknown, so that, in dealing with a problem which 

 would now be solved by the aid of several such symbols, as 

 X, y, z, etc., he was obliged to adopt some expedient, such 

 as to make mentally such combinations and arrangements 

 as to get along with only one. It is easy to see how much 

 ingenuity must often have been required to accomplish this. 

 It is a curious and surprising fact that algebraic asalysis was 

 subjected to this same limitation down to a comparatively 

 recent period. In place of the exponents at present used to 

 indicate the powers to which quantities are raised, Diophantos 

 designated the square and cube of the unknown by the initial 

 letters of the corresponding words in Greek. Thus the un- 

 known is represented by the character ?, standing for the 

 word apid/.to? (i.e., number), which is also frequently writ- 

 ten out in full ; the square of the same by 6'', a contraction 

 for Suva/ill? (power); and the cube by 7^", a contraction for 

 Hv/3o? (cube). Higher powers up to the sixth were indi- 

 cated by combination or repetition of these symbols. The 

 origin of the character for arithmos is uncertain : it may be 

 the final sigma of this word, or it may be a contraction of 

 ap, the Brst two letters of the same, or it may be derived 

 from an old Egpytian symbol for the unknown. When 

 oblique cases of these quantities are required, the words for 

 square and cube are written out in full, while the practice 

 varies with regard to arithmos, the word being sometimes 



